Radicals of a Left-Symmetric Algebra on a Nilpotent Lie Group

Chang, Kyeong-Soo; Kim, Hyuk; Lee, Hyun-Koo

doi:10.4134/bkms.2004.41.2.359

Cited by 4 publications

(7 citation statements)

References 7 publications

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“…If we believe that rad(A) is an ideal in this case, it follows that rad(A) ⊆ nil(A) and hence rad(A) = nil(A). More generally the following result is proved in [22]:…”

Section: Its Radical Is Given Bymentioning

confidence: 95%

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Left-symmetric algebras, or pre-Lie algebras in geometry and physics

2006

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Abstract. In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.

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“…If we believe that rad(A) is an ideal in this case, it follows that rad(A) ⊆ nil(A) and hence rad(A) = nil(A). More generally the following result is proved in [22]:…”

Section: Its Radical Is Given Bymentioning

confidence: 95%

“…Also it need not coincide with the Koszul radical rad(A) of A. But we have the following result [21]: Suppose that the Lie algebra g A is nilpotent. Since rad(A) is complete the Lemma implies that rad(A) is left-nilpotent.…”

Section: Its Radical Is Given Bymentioning

confidence: 97%

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

2006

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“…Now we shall briefly discuss the problem of extension of a left-symmetric algebra by another left-symmetric algebra. To our knowledge, the notion of extensions of left-symmetric algebras has been considered for the first time in [9], to which we refer the reader for more details [15].…”

Section: Recall That a Lie Algebra mentioning

confidence: 99%

Complete Left-Invariant Affine Structures on Solvable Non-Unimodular Three-Dimensional Lie Groups

K¹

2015

J Generalized Lie Theory Appl

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“…To our knowledge, the notion of extensions of left-symmetric algebras has been considered for the first time in [9], to which we refer the reader for more details. See also [4].…”

Section: Left-symmetric Algebras and Their Extensionsmentioning

confidence: 99%

Complete left-invariant affine structures on solvable non-unimodular three-dimensional Lie groups

Guediri¹,

Al-Balawi²

2014

Preprint

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In this paper, we shall use a method based on the theory of extensions of left-symmetric algebras to classify complete left-invariant affine real structures on solvable non-unimodular three-dimensional Lie groups.Recently, a method based on the theory of extensions of left-symmetric algebras has been proposed in [6] to classify complete left-invariant affine real structures on a given solvable Lie group of low dimension. Since the classification in the case of solvable unimodular Lie groups of dimension three was obtained in [5], we will use that method to carry out in this paper the classification of complete left-invariant affine structures on three-dimensional solvable non-unimodular Lie groups.The paper is organized as follows. In section 2, we will briefly recall some necessary definitions and basic results on left-symmetric algebras and their extensions. In section 3, using the classification of the threedimensional complex simple left-symmetric algebras given in [3] and a result in [10], we shall first show that

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Radicals of a Left-Symmetric Algebra on a Nilpotent Lie Group

Cited by 4 publications

References 7 publications

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

Complete Left-Invariant Affine Structures on Solvable Non-Unimodular Three-Dimensional Lie Groups

Complete left-invariant affine structures on solvable non-unimodular three-dimensional Lie groups

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